Route determination in dynamic and uncertain environments

ABSTRACT

Techniques for use in connection with determining an optimized route for a vehicle include obtaining a target state, a fixed initial position of the vehicle, and dynamic flow information, and determining an optimized route from the fixed initial position to the target state using the dynamic flow information.

RELATED APPLICATIONS

The present application claims the benefit under 35 U.S.C. § 119(e) of U.S. Provisional Patent Application Ser. No. 62/689,011 entitled “OPTIMAL SHIP ROUTING IN STRONG, DYNAMIC, AND UNCERTAIN OCEAN CURRENTS AND WAVES,” filed Jun. 22, 2018, which is incorporated herein by reference in its entirety.

FIELD OF INVENTION

The techniques described herein relate to the field of automatic determination of routes in dynamic and uncertain environments, and more particularly to stochastic techniques for determining routes that optimize one or more performance criteria.

BACKGROUND

Planning optimal routes for vehicles is important in many applications. For example, shipping routes can be optimized to reduce time and/or risk or optimized to reduce cost. Planning optimal routes can increase profits associated with a shipping business or ensure the saftety of a vehicle.

SUMMARY

Some aspects of the present application relate to a method for use in automatically determining an optimized route for a vehicle. The method includes, using at least one computer hardware processor to perform obtaining a target state, a fixed initial position of the vehicle, and dynamic flow information; and determining an optimized route from the fixed initial position to the target state using the dynamic flow information

Some aspects of the present application relate to at least one non-transitory computer-readable storage medium storing processor executable instructions that, when executed by at least one computer hardware processor, cause the at least one computer hardware processor to perform a method for use in automatically determining an optimized route for a vehicle. The method includes, using at least one computer hardware processor to perform obtaining a target state, a fixed initial position of the vehicle, and dynamic flow information; and determining an optimized route from the fixed initial position to the target state using the dynamic flow information

Some aspects of the present application relate to a A system that includes at least one computer hardware processor; and at least one non-transitory computer-readable storage medium storing processor executable instructions that, when executed by the at least one computer hardware processor, cause the at least one computer hardware processor to perform a method for use in automatically determining an optimized route for a vehicle. The method includes using at least one computer hardware processor to perform obtaining a target state, a fixed initial position of the vehicle, and dynamic flow information; and determining an optimized route from the fixed initial position to the target state using the dynamic flow information

The foregoing is a non-limiting summary of the invention, which is defined by the attached claims.

BRIEF DESCRIPTION OF DRAWINGS

Various aspects and embodiments of the disclosure provided herein are described below with reference to the following figures. It should be appreciated that the figures are not necessarily drawn to scale. Items appearing in multiple figures are indicated by the same or a similar reference number in all the figures in which they appear.

FIG. 1-1 is a block diagram of a route determination system, according to some embodiments.

FIG. 1-2 is a block diagram of a computer system, according to some embodiments.

FIG. 1-3 is a flowchart of a method of determining optimized routes, according to some embodiments.

FIG. 2-1 is a plot of a maximum speed profile produced by wind, according to some embodiments.

FIG. 2-2 is a plot of an optimal tacking path for a sailboat, according to some embodiments.

FIG. 2-3A is a map of a time evolution of optimal trajectories for two gliders after five days, according to some embodiments

FIG. 2-3B is a map of a time evolution of the optimal trajectories for two gliders after ten days, according to some embodiments

FIG. 2-3C is a map of a time evolution of the optimal trajectories for two gliders after twenty days, according to some embodiments

FIG. 2-3D is a map of a time evolution of the optimal trajectories for two gliders after twenty-five days, according to some embodiments

FIG. 2-4A is a magnified view of the time evolution of the optimal trajectory for the first glider, according to some embodiments, and corresponds to the region in the lower dashed box of FIG. 2-3B.

FIG. 2-4B is a magnified view of the time evolution of the optimal trajectory for the second glider, according to some embodiments, and corresponds to the region in the upper dashed box of FIG. 2-3B.

FIG. 2-4A is a plot of minimum arrival-times for the first glider, according to some embodiments.

FIG. 2-5B is a plot of minimum arrival-times for the second glider, according to some embodiments.

FIG. 3-1 is a schematic of a setup for determining a stochastic time-optimal path, according to some embodiments.

FIG. 3-2A is a plot of the mean of a coefficient for a dynamically orthogonal decomposition of a velocity field, according to some embodiments.

FIG. 3-2B is a plot of the mode of a coefficient for a dynamically orthogonal decomposition of a velocity field, according to some embodiments.

FIG. 3-2C is a plot of the probability density function (PDF) of a coefficient for a dynamically orthogonal (DO) decomposition of a velocity field, according to some embodiments.

FIG. 3-3 is a cumulative histogram of relative error in arrival-time determined between the DO decomposition approach and a Monte Carlo (MC) approach, according to some embodiments.

FIG. 3-4A is a Frechet distance (normalized) between reachability fronts computed using the MC approach and the DO approach at time T=25, according to some embodiments.

FIG. 3-4B is a Frechet distance (normalized) between reachability fronts computed using the MC approach and the DO approach at time T=50, according to some embodiments.

FIG. 3-4C is a Frechet distance (normalized) between reachability fronts computed using the MC approach and the DO approach at time T=75, according to some embodiments.

FIG. 3-4D is a Frechet distance (normalized) between reachability fronts computed using the MC approach and the DO approach at time T=100, according to some embodiments.

FIG. 3-5A is a plot of the spatial distribution of reachability fronts at time t=0.05, according to some embodiments.

FIG. 3-5B is a plot of the spatial distribution of reachability fronts at time t=25, according to some embodiments.

FIG. 3-5C is a plot of the spatial distribution of reachability fronts at time t=50, according to some embodiments.

FIG. 3-5D is a plot of the spatial distribution of reachability fronts at time t=75, according to some embodiments.

FIG. 3-5E is a plot of the spatial distribution of reachability fronts at time t=100, according to some embodiments.

FIG. 3-5F is a plot of the spatial distribution of reachability fronts at time t=125, according to some embodiments.

FIG. 3-6 is a plot of stochastic time-optimal paths for three targets, according to some embodiments.

FIG. 3-7 includes plots showing the mean field, variance of the DO coefficients, the DO mode fields for five mode fields, and the marginal PDF of the corresponding DO coefficient of a stochastic double-gyre flow field at a beginning time, t=0 days, and an end time, t=13.5 days, according to some embodiments.

FIG. 3-8 includes plots showing the stochastic flow field for a first realization with a negative first coefficient and a second realization with a positive first coefficient at a beginning time, t=0 days, and an end time, t=13.5 days, according to some embodiments.

FIG. 3-9 includes plots showing stochastic reachability fronts over time, according to some embodiments.

FIG. 3-10A is a plot of the stochastic time-optimal paths colored with the velocity DO coefficient 1, according to some embodiments.

FIG. 3-10B is a plot of the stochastic time-optimal paths colored with the arrival time at the target, according to some embodiments.

FIG. 3-11A includes plots of the DO mean flow (row A) and the DO mean flow for mode 1 and mode 2 fields (row B), according to some embodiments.

FIG. 3-11B includes plots of the marginal PDFs of coefficients 1 and 2 (row C), according to some embodiments.

FIG. 3-11C includes plots of the variance of the first eight modes (row D), according to some embodiments.

FIG. 3-12 includes plots of two realization of the stochastic flow fields at different times, according to some embodiments.

FIG. 3-13 includes plots of stochastic reachability fronts overlaid on the velocity streamlines of mode 1 (column A) and mode 2 (column B).

FIG. 3-14 includes plots of the stochastic time-optimal paths colored by velocity coefficient 1 (column A) and coefficient 2 (column B) to target 2, target 5, and four other targets, according to some embodiments.

FIG. 3-15 includes plots of arrival time distributions at each of six targets, according to some embodiments.

FIG. 4-1 is a schematic of a minimum-risk time-optimal path planning setup, according to some embodiments.

FIG. 4-2A is a plot of the domain of flow strength for a stochastic simulated front crossing, according to some embodiments.

FIG. 4-2B is a plot of the PDF flow strength for a stochastic simulated front crossing, according to some embodiments.

FIG. 4-3A is a plot of stochastic reachability fronts at time T=2, according to some embodiments.

FIG. 4-3B is a plot of stochastic reachability fronts at time T=2.5, according to some embodiments.

FIG. 4-3C is a plot of stochastic reachability fronts at time T=3.5, according to some embodiments.

FIG. 4-3D is a plot of stochastic reachability fronts at time T=6, according to some embodiments.

FIG. 4-3E is a plot of stochastic reachability fronts at time T=9, according to some embodiments.

FIG. 4-3F is a plot of a time-optimal path distribution for a vehicle navigating in a stochastic steady front with uncertain flow strength, according to some embodiments.

FIG. 4-4 includes plots of cost matrices (row 1), risks associated with each path (row 2), and the risk-optimal paths (row 3) for risk-seeking behavior (column a), risk-neutral behavior (column b), and risk-averse behavior (column c), according to some embodiments.

FIG. 4-5A is a plot of risk-optimal waypoint objectives for a stochastic steady front crossing, according to some embodiments.

FIG. 4-5B is a plot of PDF error due to following a risk-optimal path for a stochastic steady front crossing, according to some embodiments.

FIG. 4-6A is a plot of the Frechet distance between the time-optimal path and the risk-optimal, risk-seeking path, according to some embodiments.

FIG. 4-6B is a plot of the Frechet distance between the time-optimal path and the risk-optimal, risk-neutral path, according to some embodiments.

FIG. 4-6C is a plot of the Frechet distance between the time-optimal path and the risk-optimal, risk-averse path, according to some embodiments.

FIG. 4-7A is a plot of risk-optimal heading objectives for a stochastic steady front crossing, according to some embodiments.

FIG. 4-7B is a plot of the PDF of errors due to following the risk-optimal heading objectives for a stochastic steady front crossing, according to some embodiments.

FIG. 4-8A is a plot of a realized risk-seeking path corresponding to a particular flow realization and is colored by the discrete Frechet distance between this path and the true-optimal path for the realized environmental flow, according to some embodiments.

FIG. 4-8B is a plot of a realized risk-neutral path corresponding to a particular flow realization and is colored by the discrete Frechet distance between this path and the true-optimal path for the realized environmental flow, according to some embodiments.

FIG. 4-8C is a plot of a realized risk-averse path corresponding to a particular flow realization and is colored by the discrete Frechet distance between this path and the true-optimal path for the realized environmental flow, according to some embodiments.

FIG. 4-9 includes plots of the mean (first row), standard deviation (second row), skewness (third row), perturbation from the DO mean using a first realization (fourth row), and perturbation from the DO mean using a second realization (fifth row) for a stochastic wind-driven double-gyre flow at times T=0 days (first column), T=6.75 days (second column), and T=13.5 days (third column), according to some embodiments.

FIG. 4-10 includes plots of cost matrices (row 1 ), risks associated with each path (row 2), and the risk-optimal paths (row 3) for risk-seeking behavior (column a), risk-neutral behavior (column b), and risk-averse behavior (column c), according to some embodiments.

FIG. 4-11A is a plot of risk-optimal waypoint objectives for a stochastic double gyre flow field, according to some embodiments.

FIG. 4-11B is a plot of PDF error due to following a risk-optimal path in a stochastic double gyre flow field, according to some embodiments.

FIG. 4-12A is a plot of the Frechet distance between the time-optimal path and the risk-optimal, risk-seeking path, according to some embodiments.

FIG. 4-12B is a plot of the Frechet distance between the time-optimal path and the risk-optimal, risk-neutral path, according to some embodiments.

FIG. 4-12C is a plot of the Frechet distance between the time-optimal path and the risk-optimal, risk-averse path, according to some embodiments.

FIG. 4-13A is a plot of risk-optimal heading objectives for a stochastic double gyre flow field, according to some embodiments.

FIG. 4-13B is a plot of the PDF of errors due to following the risk-optimal heading objectives for a stochastic double gyre flow field, according to some embodiments.

FIG. 4-14A is a plot of a realized risk-seeking path corresponding to a particular flow realization and is colored by the discrete Frechet distance between this path and the true-optimal path for the realized environmental flow, according to some embodiments.

FIG. 4-14B is a plot of a realized risk-neutral path corresponding to a particular flow realization and is colored by the discrete Frechet distance between this path and the true-optimal path for the realized environmental flow, according to some embodiments

FIG. 4-14C is a plot of a realized risk-averse path corresponding to a particular flow realization and is colored by the discrete Frechet distance between this path and the true-optimal path for the realized environmental flow, according to some embodiments

FIG. 4-15A includes plots of the mean (first row), mode 1 (second row), mode 2 (third row), and mode 3 (fourth row) for a stochastic flow exiting a strait at times T=0 minutes (first column), T=50 minutes (second column), and T=100 minutes (third column), according to some embodiments.

FIG. 4-15B includes plots of the marginal PDF for the coefficient of mode 1 (first row), the marginal PDF for the coefficient of mode 2 (second row), and the marginal PDF for the coefficient of mode 3 (third row) for a stochastic flow exiting a strait at times T=0 minutes (first column), T=50 minutes (second column), and T=100 minutes (third column), according to some embodiments.

FIG. 4-16 includes plots of the standard deviation (first row), the skewness (second row), the kurtosis (third row), a first representative realization (fourth row), and a second representative realization (fifth row) of a velocity field for a stochastic flow exiting a strait, according to some embodiments.

FIG. 4-17 includes plots of stochastic reachability fronts colored by DO velocity coefficient #1 at different times, according to some embodiments.

FIG. 4-18 includes plots of time-optimal path distributions colored by arrival time, according to some embodiments.

FIG. 4-19 includes plots of risk-optimal paths (for risk-seeking, risk-neutral, and risk-averse choices) with waypoint objectives in a stochastic flow exiting a strait for five different target locations (column 1) and corresponding PDF of errors due to following the risk-optimal paths (column 2), according to some embodiments.

FIG. 4-20 includes plots of risk-optimal heading objectives for risk-seeking, risk-neutral, and risk-averse choices in a stochastic flow exiting a strait for five different target locations (column 1) and corresponding PDF of errors due to following the risk-optimal heading objectives (column 2), according to some embodiments.

FIG. 4-21 includes plots of trajectories obtained by following risk-optimal headings (for risk-seeking (first column), risk-neutral (second column), and risk-averse (third column) choices) with waypoint objectives in a stochastic flow exiting a strait for five different target locations (rows a-e), according to some embodiments.

DETAILED DESCRIPTION

Determining optimized paths has applications for exploratory, surveillance, and imaging missions using UABs and/or UAVs as well as in determining optimized shipping routes for container ships. In many application, water and/or air flows encountered by the vehicles may be comparable in magnitude to the nominal speeds of the vehicles.

The inventors have recognized and appreciated that rigorous and efficient techniques for determining optimized routes are needed. Any number of performance criteria may be optimized, such as travel time, energy consumed, or risk encountered.

The inventors have recognized and appreciated that a deductive forward computation of optimized trajectories of vehicles operating in dynamic and uncertain flows may be determined with a computational efficiency that is one or two orders of magnitude better than conventional techniques. In some embodiments, a time-optimized route to a target state is determined by starting with a fixed initial position and determining a forward reachable set of fronts using an unsteady Hamilton-Jacobi (HJ) equation.

The inventors have further recognized and appreciated combining decision theory with stochastic path-planning can result in a new partial differential equation-based scheme for risk optimal route planning in uncertain and dynamic flows. By combining a principled risk optimality criterion grounded in decision theory with stochastic dynamicallt orthogonal level-set equations, an efficient computational scheme to predict optimized paths from a distribution of stochastic optimized paths is determined.

The inventors have further recognized and appreciated that accounting for uncertainty in optimized route path planning results is important for many applications. To efficiently solve the partial differential equations involved, some embodiments use dynamically orthogonal reduced-order projections that result in several orders of magnitude in computational speed-up relative to Monte Carlo techniques.

FIG. 1 is a schematic diagram of a route determination system 1-100, according to some embodiments. The route determination system 1-100 includes a computer server 1-101, a computer 1-103, a data storage device 1-105, a first vehicle 1-107, and a second vehicle 1-109, which may be communicatively coupled together by a network 1-110. The network 1-110 comprises one or more networking devices for transmitting information from one point of the network 1-110 to another. The network 1-110 may include a local area network (LAN), a wide area network (WAN), and/or the internet. The network 1-110 may include connections, such as wired links, wireless communications links, and/or fiber optic cables. The network 1-110 may include wireless access points, switches, routers, gateways, and/or other networking equipment as well as any suitable wired and/or wireless communication medium or media for exchanging data between two or more computers, including the Internet. The wireless connections may be implemented using radio signals, optical communication signals, and/or satellite links. Computer server 1-101, computer 1-103, and data storage device 1-105 are illustrated as connected to network 1-110 with a wired connection (either electrical or optical). The first vehicle 1-107 and a second vehicle 1-109 are illustrated as being wirelessly connected to the network 1-110. However, embodiments are not so limited. For example, any of the components may be connected with a wired connection or a wireless connection.

As depicted, the first vehicle 1-107 is illustrated as a shipping vessel and a second vehicle 1-109 is illustrated as a submarine. In some embodiments, the vehicles may be other watercraft such as tankers, bulk carriers, container vessels, passenger vessels, autonomous underwater vehicles (AUVs), sailboats, underwater gliders, or yachts. In other embodiments, the vehicle need not be watercraft. For example, in some embodiments, the vehicles may include aircraft, such as airplanes, helicopters, drones, or unmanned aerial vehicles (UAVs). In some embodiments, the vehicles may be manned or unmanned. In some embodiments, the vehicles may be autonomous or manually controlled by a human operator.

The data storage device 1-105 may be one or more storage devices, such as hard drives, tape drives, optical drives, and other suitable types of devices. The data storage device 1-105 may be located in a single location or may be distributed in different locations. In some embodiments, the data storage device 1-105 may provide data and other information to the computer server 1-101, the computer 1-103, the first vehicle 1-107, or the second vehicle 1-109. For example, the data storage device 1-105 may contain information about the environment associated with one or more vehicles, such as dynamic flow information associated with the air or water. For example, the dynamic flow information may include one or more flow fields.

FIG. 1-2 is block diagram of a computing device 2-100 according to some embodiments. The computer server 1-101 or the computer 1-103 may have one or more of the components described in connection with the computing device 2-100. Additionally, the first vehicle 1-107 and/or the second vehicle 1-109 may include a computer with one or more of the components described in connection with the computing device 2-100.

Computing device 2-100 may include at least one computer hardware processor 1-210, a memory 1-220, a non-volatile storage 1-230, an input/output (I/O) device 1-240, a network adapter 1-250, and/or a display 1-260. Computing device 1-200 may be, for example, a desktop or laptop personal computer, a personal digital assistant (PDA), a smart mobile phone, a tablet computer, a computer server, or any other suitable computing device.

Network adapter 1-250 may be any suitable hardware and/or software to enable the computing device 1-200 to communicate wired and/or wirelessly with any other suitable computing device over any suitable computing network, such as network 1-100. The network adapter 1-250 may be used to obtain information from the data storage device 1-105. For example, dynamic flow information and flow uncertainty information may be obtained from the data storage device 1-105.

The display 1-260 may be any suitable display for displaying to a user a visual representation of the optimized route determination results. For example, the display 1-260 may be a computer monitor, a LCD display, or a touchscreen display. The results displayed may include a map, a list of headings, and/or instructions for following the determined route.

The non-volatile storage 1-230 may be adapted to store data to be processed and/or instructions to be executed by processor 602. For example, the non-volatile storage 1-230 may include at least one non-transitory computer-readable storage medium storing processor executable instructions that, when executed by at least one computer hardware processor, such as the processor 1-210, cause the at least one computer hardware processor to perform a method for use in automatically determining an optimized route for a vehicle.

The memory 1-220 may be a volatile memory device such as random-access memory (RAM) that may be controlled by the processor 1-210.

Computer hardware processor 1-210 enables processing of data and execution of instructions. The processor 1-210 may cause computer executable instructions stored in the non-volatile storage 1-230 to be loaded into the memory 1-220. The processor 1-210 may then the instructions to perform a method for use in automatically determining an optimized route for a vehicle.

The data and instructions stored on the non-volatile storage 1-230 may comprise computer-executable instructions implementing techniques which operate according to the principles described herein.

While not illustrated in FIG. 1-2, a computing device may additionally have one or more components and peripherals, including input and output devices. These devices can be used, among other things, to present a user interface. Examples of output devices that can be used to provide a user interface include printers or display screens for visual presentation of output and speakers or other sound generating devices for audible presentation of output. Examples of input devices that can be used for a user interface include keyboards, and pointing devices, such as mice, touch pads, and digitizing tablets. As another example, a computing device may receive input information through speech recognition or in other audible format.

FIG. 1-3 is a flowchart of a method 1-300 for determining an optimized route for a vehicle, according to some embodiments. The method 1-300 may be implemented using all or a portion of the route determination system 1-100. For example, a computer on the first vehicle 1-107 or the second vehicle 1-109 may perform all of the actions in the method 1-300. Alternatively, the server 1-101 and/or the computer 1-103 may perform a portion of the method 1-300, while other actions of the method 1-300 may be executed by the first vehicle 1-107 or the second vehicle 1-109. For example, the server 1-101 may determine an optimized route for the first vehicle 1-109 using flow information from data storage device 1-105 and then send information detailing the optimized route to the first vehicle 1-109 where a computer on the first vehicle 1-109 may control the first vehicle 1-109 to implement the optimized route.

At act 1-301, the method includes obtaining a target state, a fixed initial position, and dynamic flow information. In some embodiments, one or more computers performing the method 1-300 may perform the act of obtaining 1-301. For example, any of the computers or servers described above may perform the act of obtaining. In some embodiments, the target state includes a desired destination for the vehicle. In some embodiments, the fixed initial position may include the current position of the vehicle. For example, radar or GPS may be used to determine the initial position of the vehicle. The dynamic flow information may include, for example, a flow field. The dynamic flow information may be obtained from a third party, such as a weather service. The dynamic flow information may also include uncertainty information pertaining to the certainty that the flow field is accurate. The dynamic flow information may, for example, be stored on the data storage device 1-105 and obtained via the network 1-110.

At act 1-303, the method 1-300 includes determining an optimized route from the fixed initial position to the target state using the dynamic flow information. In some embodiments, the act of determining 1-303 may include any number of actions described in detail in Section II, Section III, and/or Section IV, below.

At act 1-305, the method 1-300 includes displaying one or more indication of the optimized route. This act 1-305 may not be performed in every embodiment. In some embodiments, the indications of the optimized route may include a list of speeds and/or heading to be implemented by a human operator of the vehicle. In other embodiments, the indications may include a map of the vehicle and its surrounding.

At act 1-307, the method 1-300 includes automatically controlling the heading of the vehicle to follow the optimized route. This act 1-307 may not be performed in every embodiment. In some embodiments, the act 1-307 is performed for an autonomous, unmanned vehicle. In other embodiments, the act 1-307 may be performed even when there is a human operator of the vehicle if, for example, the human operator is operating the vehicle in an autopilot mode.

Some aspects of the technology described herein may be understood further based on the disclosure and illustrative non-limiting embodiments provided below in Sections II, III, and IV. Each of Section II, III, and IV are self-contained such that equations referenced in each section only refer to the equation in the respective section. Similarly, sub-sections and appendices referenced in each section refer only to the sub-sections and appendices associated with the respective section. Similarly, references cited in each section refer only to the references associated with the respective section. 

What is claimed is:
 1. A method for use in automatically determining an optimized route for a vehicle, the method comprising: using at least one computer hardware processor to perform: obtaining a target state, a fixed initial position of the vehicle, and dynamic flow information; and determining an optimized route from the fixed initial position to the target state using the dynamic flow information.
 2. The method of claim 1, wherein determining the optimized path comprises calculating a forward reachability front set by numerically solving an unsteady Hamilton-Jacobi (HJ) equation.
 3. The method of claim 1, wherein determining the optimized path comprises using dynamic stochastic order reduction.
 4. The method of claim 3, wherein using dynamic stochastic order reduction comprises using dynamically orthogonal (DO) field equations and/or determining efficient stochastic DO level-set equations.
 5. The method of claim 1, wherein determining the optimized path comprises predicting a probabilistic velocity field.
 6. The method of claim 5, wherein predicting the probabilistic velocity field comprises solving discrete stochastic dynamically orthogonal (DO) barotropic quasi-geostrophic equations.
 7. The method of claim 5, wherein predicting the probabilistic velocity field comprises solving discrete stochastic dynamically orthogonal (DO) primitive equations.
 8. The method of claim 5, wherein determining the optimized path further comprises performing stochastic optimized path planning by solving stochastic DO level-set equations and computing discrete time-optimal paths and headings using backtracking equations.
 9. The method of claim 8, wherein determining the optimized path further comprises performing risk evaluation and optimization.
 10. The method of claim 9, wherein performing risk evaluation and optimization comprises: simulating trajectories using waypoint objective or heading objectives. computing an error metric matrix for the simulated trajectories computing a cost matrix for the simulated trajectories based on the error metric matrix computing a risk for each of the simulated trajectories based on the cost matrix; and determining the optimized path as the simulated trajectory associated with a lowest value of the computed risk.
 11. At least one non-transitory computer-readable storage medium storing processor executable instructions that, when executed by at least one computer hardware processor, cause the at least one computer hardware processor to perform a method for use in automatically determining an optimized route for a vehicle, the method comprising: obtaining a target state, a fixed initial position of the vehicle, and dynamic flow information; and determining an optimized route from the fixed initial position to the target state using the dynamic flow information.
 12. The at least one non-transitory computer-readable storage medium of claim 11, wherein determining the optimized path comprises using dynamic stochastic order reduction.
 13. The at least one non-transitory computer-readable storage medium of claim 12, wherein using dynamic stochastic order reduction comprises using dynamically orthogonal (DO) field equations and/or determining efficient stochastic DO level-set equations.
 14. The at least one non-transitory computer-readable storage medium of claim 12, wherein determining the optimized path further comprises: performing stochastic optimized path planning by solving stochastic DO level-set equations and computing discrete time-optimal paths and headings using backtracking equations; performing risk evaluation and optimization by: simulating trajectories using waypoint objective or heading objectives; computing an error metric matrix for the simulated trajectories; computing a cost matrix for the simulated trajectories based on the error metric matrix; and computing a risk for each of the simulated trajectories based on the cost matrix; and determining the optimized path as the simulated trajectory associated with a lowest value of the computed risk.
 15. A system, comprising: at least one computer hardware processor; and at least one non-transitory computer-readable storage medium storing processor executable instructions that, when executed by the at least one computer hardware processor, cause the at least one computer hardware processor to perform a method for use in automatically determining an optimized route for a vehicle, the method comprising: obtaining a target state, a fixed initial position of the vehicle, and dynamic flow information; and determining an optimized route from the fixed initial position to the target state using the dynamic flow information.
 16. The system of claim 15, wherein determining the optimized path comprises calculating a forward reachability front set by numerically solving an unsteady Hamilton-Jacobi (HJ) equation.
 17. The system of claim 15, wherein determining the optimized path comprises using dynamic stochastic order reduction.
 18. The system of claim 17, wherein using dynamic stochastic order reduction comprises using dynamically orthogonal (DO) field equations and/or determining efficient stochastic DO level-set equations.
 19. The system of claim 17, wherein determining the optimized path further comprises: performing stochastic optimized path planning by solving stochastic DO level-set equations and computing discrete time-optimal paths and headings using backtracking equations; performing risk evaluation and optimization; and determining the optimized path as the simulated trajectory associated with a lowest value of the computed risk.
 20. The system of claim 17, wherein performing risk evaluation and optimization comprises: simulating trajectories using waypoint objective or heading objectives; computing an error metric matrix for the simulated trajectories; computing a cost matrix for the simulated trajectories based on the error metric matrix; and computing a risk for each of the simulated trajectories based on the cost matrix. 